Modigliani–Miller propositions

What did Modigliani and Miller assume?

In 1958 Franco Modigliani and Merton Miller asked a clean question: if you hold a company’s investments fixed, does changing how it is financed change its value? They answered it under a set of idealized assumptions, then relaxed them one at a time. The result is the standard starting point for thinking about debt.

1. Homogeneous expectations. Investors agree on a given investment’s expected cash flows.
2. Perfect capital markets . No taxes, transaction costs or bankruptcy costs, and everyone has the same information.
3. Risk-free borrowing and lending. Investors can borrow and lend at the risk-free rate.
4. No agency costs. Managers always act to maximize shareholder wealth.
5. Independent decisions. Financing and investment decisions do not affect each other.

Real markets violate every one of these, which is exactly why the framework is useful: each violation can be switched on to see what it does.

Does capital structure matter?it depends on the assumptions

• No taxes Memorize perfect markets
  • Proposition I V_L = V_U
  • Proposition II r_e rises linearly; WACC constant
• Corporate taxes Memorize interest is deductible
  • Proposition I V_L = V_U + tD
  • Proposition II r_e rises more slowly; WACC falls
• Taxes plus distress costs Use it an optimal level of debt appears
  • Direct costs legal and administrative
  • Indirect costs lost customers, staff and chances

Without taxes: why does financing not matter?

Modigliani–Miller Proposition I without taxes says that the value of a levered company equals the value of an unlevered one.

$$V_L = V_U$$

Value is the present value of expected future cash flows. Changing the mix of debt and equity slices the same cash flows differently but does not change their total, so a company’s value is set by its cash flows, not by its financing. The argument rests on homemade leverage : an investor who wants more (or less) leverage than the company has can borrow (or lend) personally, at the risk-free rate, and reach the same position. Because investors can undo or create the company’s leverage themselves, they will not pay extra for it. If two firms with the same cash flows traded at different values, arbitrage would close the gap.

Go deeper Homemade leverage, step by step

Take an unlevered firm (Firm A) worth 1,000, and an otherwise identical firm (Firm B) that has debt of 500. Both generate the same operating cash flows, X. An investor wants a 10% stake in B’s leveraged equity. Instead she buys 10% of A’s shares for 100 and borrows 50 at the risk-free rate (the same rate B pays, by assumption).

Her payoff is 10% of X minus the interest on 50. A 10% stake in B’s equity pays 10% of X minus 10% of the interest on 500, which is the same thing. Her net outlay is 100 − 50 = 50, so a 10% stake in B’s equity should cost 50, which means B’s equity is worth 500 and B is worth 1,000, exactly like A. If B’s equity were worth more, she could sell it, replicate its payoffs more cheaply, and pocket a riskless profit. Trades like that push the two values together.

MM do not assume bankruptcy away. They assume it carries no cost.

Higher leverage does make equity riskier, though, which brings in the second proposition.

Modigliani–Miller Proposition II without taxes says the cost of equity rises linearly with the debt-to-equity ratio.

$$r_e = r_0 + (r_0 - r_d)\,\frac{D}{E}$$

Here $r_0$ is the cost of capital of an all-equity firm, $r_d$ the cost of debt, and $D$ and $E$ the market values of debt and equity. The intercept is $r_0$ and the slope is $r_0 - r_d$. Debt looks cheaper, but every extra dollar of it makes shareholders’ returns riskier, and they demand more. The rise in $r_e$ exactly cancels the saving from cheaper debt, so WACC stays at $r_0$.

With corporate taxes: why does debt add value?

Interest is deductible, so a profitable company that borrows pays less tax. The saving is the debt tax shield . If the debt is permanent, the present value of the shield is the tax rate times the debt, $tD$.

$$V_L = V_U + tD$$

The only reason value rises is the tax shield, and the higher the tax rate, the greater the benefit of debt. Because value rises with debt, WACC must fall with debt: debt is now cheaper still (after tax), and the higher cost of equity no longer cancels the saving. The cost of equity still rises with D/E, but more slowly, because of the $(1 - t)$ term.

$$r_e = r_0 + (r_0 - r_d)(1 - t)\,\frac{D}{E}$$

Set $t = 0$ and you get back the no-tax formula. Taken literally, the with-tax model says a company should borrow almost without limit, with 100% debt optimal. That conclusion is at odds with reality, and it comes from leaving out costs of financial distress (next section).

Why not borrow 100%? Costs of financial distress

Financial distress is the heightened uncertainty about a company’s ability to meet its obligations because earnings have weakened or losses have appeared. The cost of financial distress is what that uncertainty costs, even before a bankruptcy filing.

Cost of financial distressrises as default becomes likelier

• Direct costs Use it cash spent on bankruptcy
  • Legal and administrative fees paid out in bankruptcy
• Indirect costs Use it harder to see, often larger
  • Lost customers, suppliers, staff even before filing
  • Forgone opportunities and reputational damage
  • Agency costs of debt managers and lenders in conflict

• Direct costs are cash spent in bankruptcy: legal and administrative fees.
• Indirect costs are harder to see and often larger: customers, suppliers and employees leave; investment opportunities are lost; reputation suffers; and managers and lenders come into conflict (the agency costs of debt).

These costs are lower when assets have a ready secondary market (airlines and shipping companies can sell aircraft and vessels) and higher for technology, pharmaceutical and service firms with mostly intangible assets. The probability of distress rises with more debt, higher sales risk, higher operating leverage and lower liquidity. Bring them into the model and the tax shield stops being free, which leads to the optimal capital structure in the next lesson.

Exam tip Without taxes: leverage reshuffles risk and WACC stays put. With taxes: the tax shield tD adds value and WACC falls.